

A255938


Langton's ant walk: number of black cells on the infinite grid after the ant moves n times.


22



0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 9, 8, 7, 6, 7, 6, 7, 8, 9, 10, 9, 10, 11, 12, 13, 12, 11, 10, 9, 10, 9, 10, 11, 12, 13, 12, 13, 14, 15, 16, 15, 14, 13, 12, 13, 12, 11, 12, 13, 12, 13, 14, 15, 16, 15, 14, 13, 12, 13, 12, 13, 14, 15, 16, 15, 16, 17
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OFFSET

0,3


COMMENTS

The ant starts from a completely white grid.
After n steps, the direction in which the ant is facing is 90 degree * a(n). For each 360 degrees, the ant makes a full turn.
The ant's position after n steps is Sum_{k=1..n} e^(a(n)*i*Pi/2) when expressed as a complex number. (End)


REFERENCES

D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 63.


LINKS



FORMULA



MATHEMATICA

size = 10;
grid = SparseArray[{}, {size, size}, 1];
{X, Y, n} = {size, size, 0}/2 // Round;
While[1 <= X <= size && 1 <= Y <= size,
n += grid[[X, Y]] // Sow;
grid[[X, Y]] *= 1;
{X, Y} += {Cos[\[Pi]/2 n], Sin[\[Pi]/2 n]};
] // Reap // Last // Last // Prepend[#, 0] &


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



